L(s) = 1 | − 2-s + 1.09·3-s + 4-s − 5-s − 1.09·6-s − 3.99·7-s − 8-s − 1.80·9-s + 10-s − 0.164·11-s + 1.09·12-s + 3.99·14-s − 1.09·15-s + 16-s + 1.56·17-s + 1.80·18-s + 0.826·19-s − 20-s − 4.36·21-s + 0.164·22-s + 6.99·23-s − 1.09·24-s + 25-s − 5.25·27-s − 3.99·28-s − 5.84·29-s + 1.09·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.632·3-s + 0.5·4-s − 0.447·5-s − 0.446·6-s − 1.50·7-s − 0.353·8-s − 0.600·9-s + 0.316·10-s − 0.0495·11-s + 0.316·12-s + 1.06·14-s − 0.282·15-s + 0.250·16-s + 0.380·17-s + 0.424·18-s + 0.189·19-s − 0.223·20-s − 0.953·21-s + 0.0350·22-s + 1.45·23-s − 0.223·24-s + 0.200·25-s − 1.01·27-s − 0.754·28-s − 1.08·29-s + 0.199·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9614477974\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9614477974\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 0.164T + 11T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 0.826T + 19T^{2} \) |
| 23 | \( 1 - 6.99T + 23T^{2} \) |
| 29 | \( 1 + 5.84T + 29T^{2} \) |
| 31 | \( 1 - 0.0858T + 31T^{2} \) |
| 37 | \( 1 - 8.36T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 0.179T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.366722892777330683544614623983, −8.714634073470117790432457400070, −7.77701915647577382366805683836, −7.20976538985686188980860805571, −6.27118267098906246965723314522, −5.50566922380574670480787386301, −3.98726088945100622274549244428, −3.13804758471484373911483349103, −2.51523596228541566570292647453, −0.70368659864832008180066539387,
0.70368659864832008180066539387, 2.51523596228541566570292647453, 3.13804758471484373911483349103, 3.98726088945100622274549244428, 5.50566922380574670480787386301, 6.27118267098906246965723314522, 7.20976538985686188980860805571, 7.77701915647577382366805683836, 8.714634073470117790432457400070, 9.366722892777330683544614623983